Alternate Interior Angles

When a transversal cuts two parallel lines, the interior angles on opposite sides of it are equal. This is worth more than a memorised rule — it can be derived from two facts you already have, and seeing that derivation makes the result impossible to forget.

Concept The theorem

If a transversal cuts two parallel lines, each pair of alternate interior angles is equal.

The alternate interior pairs are

(4,\ 5)

and

(3,\ 6)

— both angles lie between the parallel lines, on opposite sides of the transversal.

Concept Co-interior angles first

Before the theorem itself, note a related fact: any two adjacent angles that together form a straight line add to 180°. This applies along the transversal and along either of the parallel lines.

Co-interior angles — the interior pair on the same side of the transversal — therefore total 180°, while alternate interior angles are equal. It is worth keeping those two outcomes distinct.

Concept Deriving the result

Two facts are enough to prove the theorem:

Corresponding angles are equal when the lines are parallel:

\angle 1 = \angle 5

\angle 2 = \angle 6

\angle 3 = \angle 7

\angle 4 = \angle 8

Vertical angles are always equal, whether or not the lines are parallel.

Now chain them together:

\angle 5 = \angle 7

because they correspond.

\angle 7 = \angle 4

because they are vertical angles.

Therefore

\angle 5 = \angle 4

The same reasoning handles the other pair:

\angle 6 = \angle 8

by correspondence, and

\angle 8 = \angle 3

as vertical angles.

Therefore

\angle 6 = \angle 3

The theorem is not an extra fact to store — it follows from two rules you already know.

Example Filling in the diagram

Two parallel lines are cut by a transversal, and $\angle 4 = 70°$ . Find $\angle 5$ and $\angle 3$ .

\angle 4

and

\angle 5

are alternate interior, so

\angle 5 = 70°

\angle 3

and

\angle 4

sit together on a straight line, so they are supplementary.

180° - 70° = 110°

⟹ angle 5 = 70° and angle 3 = 110°

Check the co-interior pair as confirmation: $\angle 3 + \angle 5 = 110° + 70° = 180°$ , exactly as it should be.

Note Mistakes to avoid

Applying the theorem when the lines are not parallel — it depends on that condition entirely.

Adding alternate interior angles to 180°; they are equal, not supplementary.

Assuming co-interior angles are equal; those are the ones that total 180°.

Choosing angles on the same side of the transversal, which makes them co-interior instead.

Using exterior angles by mistake — both angles must lie between the parallel lines.

Summary

Alternate interior angles lie between two parallel lines, on opposite sides of the transversal.
When the lines are parallel, each such pair is equal: angle 4 = angle 5 and angle 3 = angle 6.
The result follows from corresponding angles plus vertical angles, so it need not be memorised separately.
Co-interior angles, on the same side, add to 180° instead of being equal.
Adjacent angles on any straight line always total 180°, which fills in the rest of the diagram.